Compact Kähler manifolds

examples

• K3 surfaces
• Calabi-Yau manifold

cohomology theory

• compact Kähler manifold of dimension n
• Dolbeault cohomology
• $h^{p,q}=\operatorname{dim} H^{p,q}(X)$
• $h^{p,q}=h^{q,p}$
• Serre duality $h^{p,q}=h^{n-p,n-q}$

Hodge decomposition theorem

• Let $M$ be a compact Kähler manifold. Let $H^{p,q}(M)$ be the space of cohomology classes represented by a closed form of type $(p,q)$. There is a direct sum decomposition

$$ H^{m}_{dR}(M;\mathbb{C})=\bigoplus_{p+q=m}H^{p,q}(M) $$ Moreover, $H^{p,q}(M)=\overline{H^{q,p}(M)}$. In other words, $H^{m}_{dR}(M)$ carries a real Hodge structure of weight $m$.